36 Prof. Challia on the Central Motion of an Elastic Fluid. 



urged that if the small vibrations of the sether are always composite, 

 and this property explains the composition of light, there ought 

 to be some indication of the same property in aerial vibrations, 

 the iBther having been assumed to be constituted like air. I 

 believe, for reasons which follow, that such indication is given 

 by certain audible sounds, which have been named Tartini's beats, 

 from their having been first discovered, or specially noticed, by 

 that musician. These beats are to be distinguished from the beats 

 of imperfect consonances, which are heard when the ratio of the 

 periods of vibration in two series of waves is a little different 

 from the ratio that produces harmony, whereas Tartini's beats 

 are best heard when the concord is most perfect. This differ- 

 ence was well understood by Smith (' Harmonics,' Prop. XI. 

 Schol. 3), who for distinction calls the latter " flutterings," but 

 appears to have noticed them only in sounding together notes 

 the ratio of whose times of vibration was expressed by numbers 

 too high for musical harmony. When the ratio is that of two 

 low numbers, as 3 and 5, the "flutterings" of Smith become 

 " the grave harmonic " of Tartini. Recently, Professor De 

 Morgan, in the Cambridge Philosophical Transactions (vol. x. 

 part 1. p. 136), has proposed to explain the grave harmonic by 

 a certain relation of the phases of the component vibrations, thus 

 making it dependent as to degree and. quality on the manner in 

 which the aerial pulses are started. But Tartini and other 

 musicians tell us that if only the condition of perfect consonance 

 is fulfilled, the grave harmonic is always equally heard. There 

 is, therefore, still something to account for. The explanation 

 offered by the theory of aerial vibrations which I have advanced, 

 is as follows. The state of the fluid as to velocity and conden- 

 sation along a straight line of propagation in the positive direc- 

 tion, is expressed generally by the equations, 



V = «ffS=S.-^ msin — (mt—x + c) X 



the symbol S embracing as many terms as we please, having 

 different values of m, \, and c. In general, to satisfy an arbi- 

 trary disturbance, the values of X, and c must not be limited as 

 to consecutiveness. But if the disturbance be such as to pro- 

 duce a perfect consonance between two notes, the component 

 vibrations will group themselves into two sets having values of X, 

 corresponding to the notes. In each set the values of m will be 

 vei-y small and those of c may be nearly consecutive, if, as the 

 analogy of light-vibrations would lead us to expect, the number 

 of the simple vibrations be very great. The resultant of each 

 set, as is known, consists of vibrations having the same value of 

 X as the components; and the most marked effect on the ear from 



